Linear Groups, Conjugacy Growth, and Classifying Spaces for Families of Subgroups

Abstract

Given a group $G$ and a family of subgroups $\mathcal{F}$, we consider its classifying space $E_{\mathcal{F}}G$ with respect to $\mathcal{F}$. When $\mathcal{F}=\mathcal{V}\mathcal{C}yc$ is the family of virtually cyclic subgroups, Juan-Pineda and Leary conjectured that a group admits a finite model for this classifying space if and only if it is virtually cyclic. By establishing a connection to conjugacy growth we can show that this conjecture holds for linear groups. We investigate a similar question that was asked by Lück--Reich--Rognes--Varisco for the family of cyclic subgroups. Finally, we construct finitely generated groups that exhibit wild inner automorphims but which admit a model for $E_{\mathcal{V}\mathcal{C}yc}(G)$ whose 0-skeleton is finite.

minor changes, to appear in International Mathematics Research Notices